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On the commutator scaling in Hamiltonian simulation with multi-product formulas

Kaoru Mizuta

TL;DR

This work addresses the challenge of achieving efficient Hamiltonian simulation with MPF when considering system size $N$ and accuracy $\varepsilon$; prior nested-commutator bounds implied poor size-scaling due to infinite BCH convergence requirements.The authors introduce a truncated BCH framework with a truncation order $p_0(N,\varepsilon)=\lceil \log(3N/\varepsilon)\rceil$ and leverage locality via a Floquet-Magnus-inspired analysis to obtain a tighter, locality-aware error bound.They prove that MPF can attain a size-dependency comparable to Trotterization, $\mathcal{O}(N^{1/(p+1)})$, while preserving polylogarithmic scaling in $1/\varepsilon$, and they quantify the associated query cost under well-conditioned MPF constructions.The results extend to related interpolation/extrapolation strategies and time-dependent MPF, offering a rigorous route to accurate and scalable quantum simulations for both finite-range and long-range local Hamiltonians.

Abstract

A multi-product formula (MPF) is a promising approach for Hamiltonian simulation efficiently both in the system size $N$ and the inverse allowable error $1/\varepsilon$ by combining Trotterization and the linear combination of unitaries (LCU). It achieves poly-logarithmic cost in $1/\varepsilon$ like LCU [G. H. Low, V. Kliuchnikov, N. Wiebe, arXiv:1907.11679 (2019)]. The efficiency in $N$ is expected to come from the commutator scaling in Trotterization, and this appears to be confirmed by the error bound of MPF expressed by nested commutators [J. Aftab, D. An, K. Trivisa, arXiv:2403.08922 (2024)]. However, we point out that the efficiency of MPF in the system size $N$ is not exactly resolved yet in that the present error bound expressed by nested commutators is incompatible with the size-efficient complexity reflecting the commutator scaling. The problem is that $q$-fold nested commutators with arbitrarily large $q$ are involved in their requirement and error bound. The benefit of commutator scaling by locality is absent, and the cost efficient in $N$ becomes prohibited in general. In this paper, we show an alternative commutator-scaling error of MPF and derive its size-efficient cost properly inheriting the advantage in Trotterization. The requirement and the error bound in our analysis, derived by techniques from the Floquet-Magnus expansion, have a certain truncation order in the nested commutators and can fully exploit the locality. We prove that Hamiltonian simulation by MPF certainly achieves the cost whose system-size dependence is as large as Trotterization while keeping the $\mathrm{polylog}(1/\varepsilon)$-scaling like the LCU. Our results will provide improved or accurate error and cost also for various algorithms using interpolation or extrapolation of Trotterization.

On the commutator scaling in Hamiltonian simulation with multi-product formulas

TL;DR

This work addresses the challenge of achieving efficient Hamiltonian simulation with MPF when considering system size $N$ and accuracy $\varepsilon$; prior nested-commutator bounds implied poor size-scaling due to infinite BCH convergence requirements.The authors introduce a truncated BCH framework with a truncation order $p_0(N,\varepsilon)=\lceil \log(3N/\varepsilon)\rceil$ and leverage locality via a Floquet-Magnus-inspired analysis to obtain a tighter, locality-aware error bound.They prove that MPF can attain a size-dependency comparable to Trotterization, $\mathcal{O}(N^{1/(p+1)})$, while preserving polylogarithmic scaling in $1/\varepsilon$, and they quantify the associated query cost under well-conditioned MPF constructions.The results extend to related interpolation/extrapolation strategies and time-dependent MPF, offering a rigorous route to accurate and scalable quantum simulations for both finite-range and long-range local Hamiltonians.

Abstract

A multi-product formula (MPF) is a promising approach for Hamiltonian simulation efficiently both in the system size and the inverse allowable error by combining Trotterization and the linear combination of unitaries (LCU). It achieves poly-logarithmic cost in like LCU [G. H. Low, V. Kliuchnikov, N. Wiebe, arXiv:1907.11679 (2019)]. The efficiency in is expected to come from the commutator scaling in Trotterization, and this appears to be confirmed by the error bound of MPF expressed by nested commutators [J. Aftab, D. An, K. Trivisa, arXiv:2403.08922 (2024)]. However, we point out that the efficiency of MPF in the system size is not exactly resolved yet in that the present error bound expressed by nested commutators is incompatible with the size-efficient complexity reflecting the commutator scaling. The problem is that -fold nested commutators with arbitrarily large are involved in their requirement and error bound. The benefit of commutator scaling by locality is absent, and the cost efficient in becomes prohibited in general. In this paper, we show an alternative commutator-scaling error of MPF and derive its size-efficient cost properly inheriting the advantage in Trotterization. The requirement and the error bound in our analysis, derived by techniques from the Floquet-Magnus expansion, have a certain truncation order in the nested commutators and can fully exploit the locality. We prove that Hamiltonian simulation by MPF certainly achieves the cost whose system-size dependence is as large as Trotterization while keeping the -scaling like the LCU. Our results will provide improved or accurate error and cost also for various algorithms using interpolation or extrapolation of Trotterization.

Paper Structure

This paper contains 17 sections, 10 theorems, 113 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Setup and multi-product formula (MPF)
  4. Error and complexity of MPF and the problem of the existing analysis
  5. Summary of our results
  6. Modified error bound and complexity of MPF
  7. Asymptotic convergence of the BCH formula for Trotterization
  8. Error bound of MPF with commutator scaling
  9. Computational cost of the well-conditioned MPF
  10. Discussion and conclusion
  11. Application to other quantum algorithms
  12. Discussion and Summary
  13. Bounds on the nested commutators and the BCH expansions
  14. Series expansions of the subsystem Trotterization and BCH formula
  15. Series expansion of the subsystem Trotterization
  16. ...and 2 more sections

Key Result

Theorem 1

(Theorem in Ref. aftab2024-mpf) Suppose that the time $\tau$ is small enough to satisfy Then, the error of MPF is bounded by where the factor $\mu_{p,m}$ is characterized by the nested commutators as

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10